On Hopf algebras of dimension $p^n$ in characteristic $p$

Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic $p>0$. We study the general structures of $p^n$-dimensional Hopf algebras over $\mathbb{k}$ with $p^{n-1}$ group-like elements or a primitive element generating a $p^{n-1}$-dimensional Hopf subalgebra. As applications, we have proved that Hopf algebras of dimension $p^2$ over $\mathbb{k}$ are pointed or basic for $p\le 5$, and provided a list of characterizations of the Radford algebra $R(p)$. In particular, $R(p)$ is the unique nontrivial extension of $\mathbb{k}[C_p{]}^{\ast }$ by $\mathbb{k}[C_p]$, where $C_p$ is the cyclic group of order $p$. In addition, we have proved a vanishing theorem for some 2nd Sweedler cohomology group and investigated the extensions of $p$-dimensional Hopf algebras. All these extensions have been identified and shown to be pointed.

31 pages LaTeX